Table 5.9.2: Our result for Bonny NIG compared to Krichene’s (2008) Normal Inverse Gaussian (NIG) Parameters
Lambda Alpha.bar Mu Sigma Beta
Our result -0.5 1.6638 0.2972 0.9687 -0.2972
Krichene 2000m1-2003m4 -0.5 1.46 0.08 2.22 -0.08
Krichene 2003m5-2005m10 -0.5 2.68 0.29 1.69 -0.17
good in modeling skewness and heavy-tailed data. This explains why these two distributions dominated others in fitting the three datasets.
In modeling the volatility process for the three datasets we compared five probability den- sity functions in the extreme value distribution family (the Weibull, lognormal, gamma, inverse gamma and the inverse Gaussian distributions) along with the normal distribution. With sim- ilar argument as presented above, the two-parameter Weibull density function is recommended for volatility in Pennsylvania electricity futures prices and Bonny light crude oil while the gamma density function is recommended for natural gas dataset. These results show the in- ability of the Gaussian process to fit high frequency data as underscored by Mandelbrot (1963) and Fama (1965) in which both authors proposed stable distributions for modeling skewness and kurtosis. The high kurtosis in the electricity returns series of Chapter Four is hereby ad- dressed. The main attraction to the GH distributions is that they are constructed as mixtures of variance-mean normal distributions with time varying stochastic variance.
Chapter 6 Conclusion
We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole lot of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things. - B. Øksendal (2000)
This Thesis is in two parts: mathematical and statistical− with each part comprising two chapters. In Part One we discussed the theory and analysis of partial differential equations using the Lie symmetry technique to analyse an evolution partial differential equation arising from financial mathematics, see for instance, equation (3.1.1) of Chapter Three. The second part concerns applications to real life problems where calibrations and statistical goodness-of-fit tests were performed.
A formula (proposition 2.3.1) for the nth prolongation of a generator Γ with k indepen- dent and p dependent variables of an nth-order partial differential equation is proposed and we claim to have extended the result (equation (2.3.6)) derived by Mahomed and Leach in 1990. The basic problem in the modeling of physical and other phenomena is to find solu- tions of differential equations. Many methods of solution of differential equations use a change of variables that transforms a given differential equation into another equation with known properties. We constructed a transformation that maps symmetries of our PDE invertibly into the heat equation which is a well studied equation with appealing characteristics. As a result
of the symmetry analysis performed we also show that our given partial differential equation admits a finite number of Lie point symmetries characterized by the six-dimensional algebra isomorphic to{sl(2, R)⊕W3} ⊕s∞A1,with one solution symmetry where the subalgebra is of the Heisenberg-Weyl type. Two general solutions calculated from the twelve optimal systems of invariant solutions are given in equations (3.3.40) and (3.3.49). It is our thinking that these equations will one day be found useful for practical applications.
The complete probability space (Ω,F,P) with natural filtration {Fk}∞1 and Levy proceses Lt were assumed. We propose a dynamic linear model (DLM) with switching regimes for modeling the stochastic volatility of log return series for prices of electricity contracts. A modified Kalman filter algorithm was introduced to fit the regime-switching Markov model and estimation of the parameters using quasi-maximum likelihood method were performed. Results displayed in Ta- ble 4.7.1 are comparable to results obtained by Kellerhals (2004) using affine structure models for spot and futures prices, and Krichene (2008) for crude oil prices using GARCH(1,1) models.
It will be of interest to compare our model with models used by Kellerhals and Krichene using our dataset.
Two observations were immediate. The first is that both small and large changes come clus- tered, i.e., there are periods of low and high volatility. The second is that, from time to time, we observe rather large changes which may be hard to reconcile with the standard distributional assumption in statistics and econometrics, that is, normality. From empirical results the dataset exhibited volatility clustering followed by mean reversion with half-life of nine months. This informed our use of Gaussian mixtures in the model. The mixing of Gaussian distributions is well suited for financial modeling, as it allows for the construction of very flexible distributions.
This fact is demonstrated in Chapter Five, where the normal-mean-variance mixture, on which the generalized hyperbolic distribution (GH) of Barndorff-Nielsen (1977) is based, generally exhibits heavier tails than the Gaussian distribution. We used this to great advantage. Inter- estingly we derived our DLM based on this idea and the generalization of the Vasicek and CIR models (or for some authors, the general Heston model) governed by the stochastic differential equation (4.3.12). The adequacy of our model was authenticated by the preliminary study of the dataset that displayed evidence of first-order autocorrelation showing that the state variable
is a first-order autoregressive AR(1) process.
The major concern in Chapter Five was the identification of the probability distribution of the process that generated the dataset. From each of the three datasets (Daily Pennsylvania Electricity Future Contract, Weekly Bonny Crude oil Spot prices and Daily Natural Gas Prices) we generated and studied two concomitant variables: log return series and volatility series.
Five probability density functions of the generalized hyperbolic family (Generalized Hyperbolic (GH), Hyperbolic (HYP), Normal Inverse Gaussian (NIG), Variance-Gamma (VG), and Skew Student-t (SSt)) and five from the extreme value family (Weibull, Gamma, Lognormal, Inverse Gaussian and Inverse gamma) were fitted to the datasets and compared with the Normal distribution as the benchmark.
We established that energy return series is fat-tailed and with significant kurtosis. The normal distribution showed very poor fit in both series. Using the Akaike Information (AIC) and the Log-likelihood (LLH) criteria, we conclude that NIG (which is a mixture of the normal and the inverse Gaussian distributions) is best suited to fit and for prediction of prices for Pennsylvania electricity future contracts. This model performed well in fitting the crude oil dataset but ranked second only to SSt. The SSt (which also has a convolution property) dominated other five candidate models (74% dominance) in the natural gas dataset. With this result we posit that SSt is good for fitting oil and gas datasets while NIG is the choice for electricity series, see Table 6.0.1. These results are not surprising. The SSt has one heavy and one semi-heavy tail, i.e., one tail determined by a polynomial and the other by an exponential behaviour. The normal inverse Gaussian processLt is a Levy process where increments inLt are distributed according to the NIG distribution. Another appeal of the NIG distribution is that it is characterised by the first four moments(mean,variance, skewness and kurtosis). These are the moments we care about for inference in real life applications including risk management and derivative pricing.
In fitting stochastic volatility series, the Weibull distribution performed wonderfully well in both the electricity and crude oil datasets while the gamma distribution is recommended for natural gas volatility series. The gamma process can be expressed as a limiting case of the generalized inverse Gaussian(GIG) process with λ = −0.5 (similar to the NIG). The Weibull on the other hand is popular in the analysis of lifetime data.
Table 6.0.1: Performance summary of different models Recommended Models
Dataset Return Series Volatility Series Electricity
NIG (72%) Weibull GH (27%)
Crude Oil SSt (32%) Weibull NIG (28%)
Natural Gas SSt (74%) Gamma GH (26%)
Contrary to the assumption of “all things being equal” there are very high probability of higher or lower energy prices than previously expected over time. This suggests that compared to the normal distribution, the actual probability distribution of return and volatility series are fat-tailed, implying that the probability of large differences in prices of energy contracts is much higher than would be implied by time-invariant unconditional Gaussian distribution.
Bibliography
[1] Aas, K. and Haff, I. H. (2006) The generalized hyperbolic skew Student’s t-distribution.
Journal of Financial Econometrics, 4 (2), 275–309.
[2] Abraham-Shrauner, B. and Guo, A. (1992) Hidden symmetries associated with the pro- jective group of nonlinear first-order ordinary differential equations. Journal of Physics, A: Mathematical and General, 25, 5597–5608.
[3] Abraham-Shrauner, B. and Guo, A. (1993) Hidden and nonlocal symmetries of nonlinear differential equations. InModern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics. Ibragimov, N. H., Torrissi, M. and Valenti, A., editors, p. 1–5. Kluwer Academics, Dordrencht.
[4] Abraham-Shrauner, B. and Guo, A. (1994) Hidden symmetries of differential equations.
Contemporary Mathematics, 160, 1–13.
[5] Abraham-Shrauner, B. and Govinder, K. S. (2006) Provenance of Type II hidden sym- metries from nonlinear partial differential equations. Journal of Nonlinear Mathematical Physics, 13, 612–622.
[6] Abraham-Shrauner, B., Govinder, K, S. and Leach, P. G. L. (1995) Integration of second- order equations not possessing point symmetries. Physics Letters A, 203, 169–174.
[7] Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. Dover Publications, New York.
[8] Akaike, H. (1973) Information theory and an extension of the maximum likelihood prin- ciple. In B. N. Petrov and F. Csaki, eds., 2nd International Symposium on Information Theory. Akademia Kiado, Budapast, 267–281.
[9] Akaike, H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, 716–723.
[10] Albrecher, H. , Mayer, P., Schoutens, W. and Tistaert, J. (2007) The little Heston trap.
WILMOT, 83–92.
[11] Alles, L. A. and Kling, J. L. (1994) Regularities in the variation of skewness in asset returns. The Journal of Financial Research,17, 427–438.
[12] Amaral, L.A.N., Plerou, V., Gopikrishnan, P., Meyer, M. and Stanley, H. E. (2000) The distribution of returns of stock prices. International Journal of Theoretical and Applied Finance, 3 (3), 365–369.
[13] Anco, S. C. and Bluman, B. W. (1996) Derivation of conservation laws from nonlocal symmetries of differential equations. Journal of Mathematical Physics,37, 2361–2375.
[14] Anderson, B. O. D. and Moore, J. B. (1979) Optimal filtering. Prentice Hall Inc., New Jersey.
[15] Andriopoulos, K. (2008) Classification of Differential Equations by Group Theoretical Methods. PhD Thesis, School of Mathematical Sciences, University of KwaZulu-Natal, Durban, Republic of South Africa.
[16] Arrigo, D. J., Broadbridge, P. and Hill, J. M. (1993) Nonclassical symmetries and the methods of Bluman-Cole and Clarkson-Kruskal. Journal of Mathematical Physics, 34, 4692–4703.
[17] Arrigo, D. J., Hill, J. M. and Broadbridge, P. (1994) Nonclassical symmetry reductions of the diffusion equation with nonlinear source. IMA Journal of Applied Mathematics, 52, 1–24.
[18] Atkinson, A. C. (1987)Plots, Transformations, and Regression.Clarendon Press, London.
[19] Azzalini, A. and Capitanio, A. (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. Journal of the Royal Statistical Society Series B, 65, 579–602.
[20] Babs, S. H. and Nowman, K. B. (1999) Kalman filtering of generalized Vasicek term structure models. Journal of Financial and Quantitative Analysis, 34(1), 115–130.
[21] Badrinath, S. G. and Chatterjee, S. (1988) On measuring skewness and elongation in common stock return distributions: the case of the market index. Journal of Business, 61(4), 451–472.
[22] Badrinath, S. G. and Chatterjee, S. (1991) A data-analytic look at skewness and elongation in common-stock-return distributions. Journal of Business and Economic Statistics,9(2), 223–233.
[23] Ball, C. A. and Torous, W. N. (1999) The stochastic volatility of short-term interest rates:
some international evidence. Journal of Finance,54(6), 2339–2359.
[24] Barndorff-Nielsen, O. E. (1977) Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society of London, A, 353, 401–419.
[25] Barndorff-Nielsen, O. E (1994) Gaussian-inverse Gaussian process and the modeling of stock returns. Paper presented at the second Workshop on Stochastics and Finance 1994 in Berlin.
[26] Barndorff-Nielsen, O. E. and Prause, K. (2001) Apparent scaling.Finance and Stochastics.
5, 103–113.
[27] Barndorff-Nielsen, O. E., Kent, J. and Sorensen, M. (1982) Normal variance-mean mixtures and z-distributions. International Statistical Review, 50, 145–159.
[28] Barndorff-Nielsen, O. E and Stelzer, R. (2005) Absolute moments of generalized hyper- bolic distributions and approximate scaling of normal inverse Gaussian L´evy processes.
Scandinavian Journal of Statistics, 32, 617–637.
[29] Benth, F. E. and Saltyte-Benth, J. (2004) The normal inverse Gaussian distribution and spot price modeling in energy markets. International Journal of Theoretical and Applied Finance, 7(2), 177–192.
[30] Birkhoff, G. D. (1933) Aesthetic Measure. Harvard University Press, Cambridge.
[31] Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy, 81, 631–659.
[32] Bluman, G. W. (1980) On the transformation of diffusion processes into the Wiener process.
SIAM Journal of Applied Mathematics,39, 238–247.
[33] Bluman, G. W. (1983) On mapping linear partial differential equations to constant coeffi- cient equations. SIAM Journal of Applied Mathematics,43(6), 1259–1273.
[34] Bluman, G. W. and Anco, S. C. (2002)Symmetry and Integration Methods for Differential Equations. Applied Mathematical Sciences no. 154. Springer-Verlag, New York.
[35] Bluman, G. W. and Cole, J. D. (1974) Similarity Methods for Differential Equations.
Springer, New York.
[36] Bluman, G. W. and Kumei, S. (1989) Symmetries and Differential Equations. Applied Mathematical Sciences no. 81, Springer-Verlag, New York.
[37] Bølviken, E. and Benth, F. E. (2000) Quantification of risk in Norwegian stocks via the normal inverse Gaussian distribution. In Proceedings of the AFIR 2000 Colloquium, Troms, Norway, 87–98.
[38] Bozdogan, H. (1987) Model selection and Akaike information criterion (AIC): the general theory and its analytical extensions. Psychometrika, 52, 345–370.
[39] Bozdogan, H. (2000) Akaikes information criterion and recent developments in the infor- mation complexity. Journal of Mathematical Psychology, 44, 62–91.
[40] Branco, M. D. and Dey, D. K. (2001) A general class of multivariate skew-elliptical distri- butions.Journal of Multivariate Analysis, 79, 99–113.
[41] Breymann, W. and Luethi, D. (2008) The ghyp: a package on generalized hyperbolic distributions.
[42] Burnham, K. P. and Anderson, D. R. (2002) Model Selection and Inference: a Practical Information-Theoretic Approach. Second Edition, Springer-Verlag, New York.
[43] Cantwell, B. J. (2002) Introduction to Symmetry Analysis. Cambridge University Press, Cambridge.
[44] C¸ etin, M. C. and Erar, A. (2002) Variable selection with Akaike information criterion: a comparative study. Hacettepe Journal of Mathematics Statistics. 31, 89–97.
[45] Champagne, B., Hereman, W. and Winternitz, P. (1991) The computer calculation of Lie point symmetries of large systems of differential equations. Computer Physics Communi- cations,66, 319–340.
[46] Chan, K. C., Karolyi, A., Longstaff, F. and Sanders, A. (1992) An empirical comparison of alternative models of the short term interest rate. Journal of Finance, 47, 1209–1227.
[47] Chen, R. R. and Scott, L. (2003) Multi-factor Cox-Ingersoll-Ross models of term structure:
estimation and tests from a Kalman filter model. Journal of Real Estate Finance and Economics, 27(2), 143–172.
[48] Chetterjee, R. (2003) Application of the Kalman filter for estimating continuous time term structure models: evidence from the UK and Germany. Working paper, Department of Economics, University of Glasgow.
[49] Chhikara, R. S. and Folks, J. L. (1989) The Inverse Gaussian Distribution. Marcel Dekker, New York.
[50] Chou, K. S and Li, G. X. (2001) A note on optimal system for the heat equation. Journal of Mathematical Analysis and Applications, 261, 741–751.
[51] Clarkson, P. A. (1989) New similarity solutions of the modified Boussinesq equation.Jour- nal of Physics A: Mathematical and General, 22, 2355–2367.
[52] Clarkson, P. A. (1989) New similarity solutions and Painlev´e analysis for the symmetric regularized long wave and the modified Benjamin-Bona-Mahoney equations. Journal of Physics A: Mathematical and General, 22, 3821–3848.
[53] Clarkson, P. A. (1995) Nonclassical symmetry reductions of the Boussinesq equation.
Chaos, Solitons and Fractals, 5, 2261–2301.
[54] Clarkson, P. A. and Kruskal, M. D. (1989) New similarity solutions of the Boussinesq equation, Journal of Mathematical Physics 30, 2201–2213.
[55] Clewlow, L. and Strickland, C. (2000)Energy Derivatives: Pricing and Risk Management.
Lacima Publications, London.
[56] Cont, R. (2001) Empirical properties of asset returns: stylized facts and statistical issues.
Quantitative Finance, 1, 223–236.
[57] Cook, R. D. and Weisberg, S. (1982)Residuals and Influence in Regression. Chapman and Hall, London.
[58] Cox, D. R. (1962) Further results on tests of separate families of hypothesis. Journal of the Royal Statistical Society, Series B,24, 406–424.
[59] Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985) The theory of the term structure of interest rates. Econometrica, 53, 385–407.
[60] Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977) Maximum likelihood from incom- plete data via EM algorithm (with discussions). Journal of the Royal Statistical Society, Series B,35, 1–38.
[61] Derman, C., Gleser, J. and Olkin, I. (1973)A Guide to Probability Theory and Applications.
Holt, Rinehart and Winston Inc., New York.
[62] Dimas, S. and Tsoubelis, D. (2005) SYM: A new symmetry-finding package for Mathe- matica. In: N. H. Ibragimov, C. Sophocleous and P. A. Damianou (eds), Group Analysis of Differential Equations, p. 64–70. University of Cyprus, Nicosia.
[63] Dimas, S. and Tsoubelis, D. (2006) A new Mathematica-based program for solving overde- termined systems of PDEs. 8th International Mathematical Symposium (Avignon, France).
[64] Dimas, S., Andriopoulos, K., Tsoubelis, D. and Leach, P.G.L. (2009) Compete specifica- tion of some partial differential equations that arise in financial mathematics. Journal of Nonlinear Mathematical Physics,16 (Supplement), 73–92.
[65] Dineen, S. (2005) Probability Theory in Finance: a Mathematical Guide to Black-Scholes Formula. Graduate Studies in Mathematics, Volume70. American Mathematical Society, Providence.
[66] Do, B. (2008) Estimating stochastic volatility option pricing models with Kalman filtering.
Working Paper. Department of Accounting and Finance, Monash University, Austrialia.
[67] Drapper, N. R. and Smith, H. (2005) Applied Regression Analysis. Fourth edition, John Wiley, New York.
[68] Dresner, L. (1999) Applications of Lie Theory of Ordinary and Partial Differential Equa- tions. IOP Publishing Ltd, Philadelphia.
[69] Duertz, D. (2007) The fBasics Package
[70] Eberlin, E. and Keller, U. (1995) Hyperbolic distributions in finance. Bernoulli, 1(3), 281–299.
[71] Etheridge, A. (2002) A Course in Financial Calculus. Cambridge University Press, Cam- bridge.
[72] Eydeland, A. and Wolyniec, K. (2003) Energy and Power Risk Management. John Wiley and Sons, New York.
[73] Fama, E. (1963) Mandelbrot and the stable Paretian hypothesis.Journal of Business, 36, 420–429.
[74] Fama, E. (1965) The behaviour of stock market prices. Journal of Business. 38, 34–105.
[75] Fernandez, C. and Steel, M. (1998) On Bayesian modeling of fat tails and skewness.Journal of the American Statistical Association, 93, 359–371.
[76] Fearn, D. H. and Nebenzahl, E. (1991) On the maximum likelihood ratio method of de- ciding between Weibull and gamma distributions. Communications in Statistics, Theory and Methods, 20(2), 579–593.
[77] Feller, W. (1957) An Introduction to Probability Theory and its Applications. Volume I, 2nd edition. John Wiley and Sons, Inc., New York.
[78] Fielitz, B. and Rozell, J. (1983) Stable distributions and mixture of distributions hy- potheses for common stock returns. Journal of the American Statistical Association, 69, 108–113.
[79] Fushchych, W. I. and Kornyak, V. V. (2001) Computer algebra application for determin- ing Lie and Lie-Backlund symmetries of differential equations. W. I. Fushchych Scientific Works, 3, 497–505.
[80] Gazizov, R. K. and Ibragimov, N. H. (1998) Lie symmetry analysis of differential equations in finance. Nonlinear Dynamics, 17, 387–407.
[81] Geyer, A. L. J. and Pichler, S. (1999) A state-space approach to estimate and test multifac- tor Cox-Ingersoll-Ross models of the term structure of interest rates. Journal of Financial Research, 22(1), 107–130.
[82] Goard, J. (2000) New solutions to the bond-pricing equation via Lie’s classical method.
Mathematical and Computer Modeling, 32, 299–313.
[83] Goldfield, S. M. and Quandt, R. E. (1973) A Markov model for switching regressions.
Journal of Econometrics, 1, 3–16.
[84] Govinder, K. S. and Leach, P.G.L. (1996) The nature and use of symmetries of ordinary differential equations. South African Journal of Science, 92, 23–28.
[85] Guillaume, D. M., Dacorogna, M. M., Dav´e , R. D., M¨uler, U. A., Oslen, R. B., and Pictet, O. V. (1997) From the bird’s eye to the microscope: a survey of new stylized facts of the intra-daily foreign exchange markets. Finance and Stochastics, 1, 95–129.
[86] Hamilton, J. D. (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57, 357–384.
[87] Hamilton, J. D. (1994)Time Series Analysis. Princeton University Press, Princeton.
[88] Hamilton, J. D. (2005)Regime-switching models. Prepared for Palgrave Dictionary of Eco- nomics. Department of Economics, University of California, San Diego.
[89] Hansen, B. (1994) Autoregressive conditional density estimation. International Economic Review, 35, 705–730.
[90] Harvey, A. C. (1989) Forecasting, Structural Time Series Models and the Kalman Filter.
Cambridge University Press, Cambridge.
[91] Harvey, A. C., Ruiz, E. and Shephard, N. (1994) Multivariate stochastic variance models.
Review of Economic Studies, 61, 247–264.
[92] Head, A. K. (1993) LIE, a PC program for Lie analysis of differential equations.Computer Physics Communications. 71, 241–248.
[93] Head, A.K. (1996) LIE, a PC program for Lie analysis of differential equations (updated version). Computer Physics Communications. 76, 311–313.
[94] Hereman, W. (1994) Review of symbolic software for the computation of Lie symmetries of differential equations. Euromathematics Bulletin, 1(1), 45–79.
[95] Hereman, W. (1996) Symbolic software for the Lie analysis. In: N. H. Ibragimov (ed), CRC Handbook of Lie Group Analysis of Differential Equations Vol. 3: New Trends in Theoretical Development and Computational Methods, p. 367–413. CRC Press, Boca Raton.
[96] Hereman, W. and Nuseir, A. (1997) Symbolic methods to construct exact solutions of nonlinear partial differential equations. Mathematics and Computers in Simulation, 43, 13–27.
[97] Heston, S. L. (1993) A closed-form solution for options with stochastic volatility, with applications to bond and currency options. Review of Financial Studies, 6, 327–343.
[98] Hull, J. (2002) Options, Futures, and Other Derivatives. 5th Edition, Prentice Hall, New Jersey.
[99] Hull, J. and White, A. (1987) The pricing of options on assets with stochastic volatilities.
Journal of Finance, 42, 281–300.
[100] Hull, J. and White, A. (1990) Pricing interest rate derivatives.Review of financial Studies, 3, 573–592.
[101] Hurst, S. R., Platen, E., and Rachev, S. T. (1995) Option pricing for asset returns driven by subordinated processes. Working Paper, The Australian National University, Canberra.
[102] Hydon, P. E. (2000)Symmetry Methods for Differential Equations: A Beginner’s Guide.
Cambridge University Press, Cambridge.
[103] Ibragimov, N. H. (1995) Galilian principle in thermal diffusion. Symmetri,2(3), 56–62.
[104] Ibragimov, N. H. (1999) Elementary Lie Group Analysis and Ordinary Differential Equa- tions. John Wiley, Chichester.
[105] Ibragimov, N. H. and Gazizov, R. K. (1998) Lie symmetry analysis of differential equations in finance. Nonlinear Dynamics, 17, 387–407.
[106] Ibragimov, N. H., Torrisi, M., Valenti, A. (1991) Preliminary group classification of equa- tions utt =f(x, νx) +g(x, νx). Journal of Mathematical Physics, 32(11), 2988–2995.
[107] Ibragimov, N. H. (2009) Selected Works IV. ALGA Publications, Karlskrona. Paper 6.
[108] Itˆo, K. (1951) On stochastic differential equations.Memoirs of the American Mathemat- ical Society, 4, 1–51.